Central Binomial Coefficients Research Articles

Solution to a problem involving central binomial coefficients

In this paper, we solve an open problem considered by Steven Finch (Central Binomial Coefficients, 2007, Available from: http://www.people.fas.harvard.edu/sfinch/csolve/cbc.pdf, p. 5), as far back as 2007, concerning the calculation of a series involving the central binomial coefficient.

On a conjecture of Graham on the p$p$‐divisibility of central binomial coefficients

On a conjecture of Graham on the p$p$‐divisibility of central binomial coefficients

Certain logarithmic integrals and associated Euler sums

We evaluate the following family of logarithmic integrals in closed form:12π∫02π|1+reit|2nlogm⁡|1+reit|dt for n∈Z⩾0 and m=0,1,2,3,4, r∈[−1,1]. In 1995, Shen conducted evaluations for specific instances where r equals −1 with n equal to 0, as well as when m assumes the values 1, 2, 3, and 4. Moreover, by employing these integrals, we derive generating functions for certain sequences that incorporate harmonic numbers and binomial coefficients. Furthermore, we assess a selection of series that encompass squared central binomial coefficients and harmonic numbers.

On a higher-order version of a formula due to Ramanujan

A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1 + 1 5 ( 1 2 ) 2 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 2 + ⋯ = Γ 4 ( 1 4 ) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1 + 1 5 ( 1 2 ) 4 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 4 + ⋯ = Γ 8 ( 1 4 ) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.

Summation Formulas on Harmonic Numbers and Five Central Binomial Coefficients

Summation Formulas on Harmonic Numbers and Five Central Binomial Coefficients

Apéry-type series and colored multiple zeta values

In this paper, we study new classes of Apéry-type series involving the central binomial coefficients and the multiple t-harmonic sums by combining the methods of iterated integrals and Fourier–Legendre series expansions, where the multiple t-harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Apéry-type series involving harmonic sums to those with products of multiple harmonic sums and multiple t-harmonic sums. We show that these series can be expressed as either the real or the imaginary part of a Q-linear combination of colored multiple zeta values of level 4. Hopefully, these relations will shed some new lights on their properties which may lead to novel approaches to irrationality questions on the Riemmann zeta values, or more generally, the multiple zeta values.

INFINITE SERIES CONCERNING HARMONIC NUMBERS AND QUINTIC CENTRAL BINOMIAL COEFFICIENTS

AbstractBy examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun [‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7]. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.

On Several Results Associated with the Apéry-like Series

In 1979, Apéry proved the irrationality of ζ(2) and ζ(3). Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions.

Some combinatorial identities containing central binomial coefficients or Catalan numbers*

In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers. The instant proof simplifies discussions in the journal papers: College Math. J. 43 (2012), no. 2, 141–146; Amer. Math. Monthly 121 (2014), no. 3, 267–267; Amer. Math. Monthly 123 (2016), no. 4, 405–406; Elem. Math. 71 (2016), no. 3, 109–121; and Mathematics 5 (2017), no. 3, Article 40, 31 pages.

Integral Representations of Catalan Numbers and Sums Involving Central Binomial Coefficients

Integral Representations of Catalan Numbers and Sums Involving Central Binomial Coefficients

Dirichlet series and series with Stirling numbers

This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.

Integral representations and properties of several finite sums containing central binomial coefficients

Integral representations and properties of several finite sums containing central binomial coefficients

Infinite series about harmonic numbers inspired by Ramanujan–like formulae

<abstract><p>By employing the coefficient extraction method from hypergeometric series, we shall establish numerous closed form evaluations for infinite series containing central binomial coefficients and harmonic numbers, including several conjectured ones made by Z.-W. Sun.</p></abstract>

On certain congruences involving central binomial coefficients

Let p be an odd prime. In this paper, using some properties of Fibonacci numbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, we establish some congruences involving central binomial coefficient modulo p and p^2. We also give some new identities for hyperbolic functions.

The integer sequence transform a → b, where bn is the number of real roots of the polynomial a 0 + a 1 x + a 2 x 2 + · · · + anxn

We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a 0 + a 1 x + a 2 x 2 + · · · + anxn . It is shown that several sequences a give the trivial sequence b = (0, 1, 0, 1, 0, 1, .), i.e., bn = n mod 2, among them the Catalan numbers, central binomial coefficients, n! and for a fixed k. We also look at some sequences a for which b is more interesting such as an = (n + 1) k for k ≥ 3. Further, general procedures are given for constructing real sequences an for which bn is either always maximal or minimal.

Sums of Pell/Lucas Polynomials and Fibonacci/Lucas Numbers

Seven infinite series involving two free variables and central binomial coefficients (in denominators) are explicitly evaluated in closed form. Several identities regarding Pell/Lucas polynomials and Fibonacci/Lucas numbers are presented as consequences.

A short derivation of an elegant sum involving central binomial coefficients due to Laszlo via a hypergeometric series approach

A short derivation of an elegant sum involving central binomial coefficients due to Laszlo via a hypergeometric series approach

From Madhava–Leibniz to Lehmer’s Limit

In a 1985 article in this Monthly, D. H. Lehmer considered a fascinating family of infinite series involving the central binomial coefficient, which exhibited limiting behavior involving the ubiquitous constant π in two different ways; yet it was not until 2012 that Dyson et al. gave a proof of the latter, naming it Lehmer’s limit in his honor. Here we give an elementary demonstration that both phenomena derive from simple transformations of the celebrated Madhava–Leibniz series for .

Explicit identities for infinite families of series involving squared binomial coefficients

Ramanujan famously evaluated the integer/half-integer moments of the complete elliptic integral of the first kind as finite sums, and, quite recently, innovative coefficient-extraction techniques based on Gauss' hypergeometric identity and a Watson–Whipple-like F23-series were introduced by Wang and Chu to generalize this hypergeometric identity due to Ramanujan, providing explicit evaluations, as finite sums, for infinite families of series involving harmonic numbers and squared central binomial coefficients. In this article, we make use of a modified Abel summation lemma to greatly build upon these results, solving two open problems concerning the recent work of Wang and Chu. We also consider how our results may be extended via fractional sums, and we introduce formulas involving the Landau constants.

Integral representations, monotonic properties, and inequalities on extended central binomial coefficient

In the paper, with the help of Kazarinoff's integral representation for the ratio of two gamma functions, with the aid the duplication formula of the digamma function, by virtue of integral representations of polygamma functions, and in the light of the L'Hôpital type monotonicity rule, the author describes extended binomial coefficients in terms of the gamma functions and the falling factorials, collects three integral representations of central binomial coefficients, establishes three integral representations of extended central binomial coefficients, proves decreasing and increasing properties of two power‐exponential functions involving extended (central) binomial coefficients, and derives several double and sharp inequalities for bounding extended (central) binomial coefficients, and compares these inequalities with known ones.